"College-level" problems

• Session 3: Models for multiplication -- use the questions on this page to develop a cohesive, comparative explanation of the set and array models for multiplication and the relationship between them.
• Session 5: Numeration Systems -- use the questions on this page to develop a cohesive, comparative explanation of the Roman and Babylonian systems in particular, and symbol value and place value systems in general.
• Session 5: Zeroes and Ones -- use the questions on this page as well as the class discussion and responses to explain options for designing numeration systems using only two meaningful characters (including true binary), and compare their properties.
• Session 6: Sticks and Stones -- use the problems on this page to develop a cohesive explanation of the Mayan system, including a comparative analysis (relative to the Hindu-Arabic system).
• Session 7: Introduction to Bases -- use the summative questions at the end of this activity to give clear sets of instructions for conversion to base $n$ using concrete [proportional] and symbolic representations, and compare the two. Of course, you may use some of the problems in this activity as examples, but be sure your instructions are stated in purely general terms.
• Sessions 8-14: Mayan addition and subtraction, Driving in Septobasiland, Regrouping in Other Bases, Mayan multiplication, Making Groups in Other Bases, Money in Septobasiland, Mayan division, Division in Other Bases, Arithmetic in Other Bases -- Use any one of these activities as a basis for developing a general explanation of the traditional algorithm for performing any one of the four arithmetic operations (choose one!) in terms of a general base (i.e., not ten). Be careful to couch your general set of instructions in terms of a general base, and not any specific one (although of course all your examples or illustrations will use particular bases).
• Session 15: Russian Peasant algorithm -- Although the two-problem paper is due before Session 15, some participants may be interested in analyzing this intriguing alternative algorithm for multiplication. Work through the questions in the handout, and write a paper which (a) briefly explains the steps in the algorithm, (b) explains mathematically why the algorithm is correct, (c) describes the mathematical skills needed and not needed, and (d) uses these to speculate why a person might prefer this algorithm over the traditional (partial products) multidigit multiplication algorithm.

Links to related sites

• learner.org's online courses "Learning Math" include Number and Operation

Links for specific class meetings

• Session 1
  • DMI sample interview write-up (from BST)
• Session 2
  • Video on the need to decompose numbers (from learner.org)
• Session 5
  • Numeration Systems
  • pp.104-108 of Bassarear's text, from Google books
  • A history of the Sumerian/Babylonian system
  • Zero in the Babylonian system
• Session 7
  • Mini-cases (a) and (b)
  • Mini-cases (c) and (d)
• Session 9
  • Base four addition applet (from learner.org)
  • reading: "Do We Really Want to Keep the Traditional Algorithms for Whole Numbers?" by John A. Van de Walle
• Session 11
  • reading: "Children's invention of multidigit multiplication and division algorithms" by Ambrose, Baek, and Carpenter
  • the multiplication half of this article made simple: "Children's mathematical understanding and invented strategies for multidigit multiplication" by Baek
• Session 14
  • reading: alternative division algorithms